# Analytical dynamics

Classical mechanics |
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Core topics |

In classical mechanics, **analytical dynamics**, or more briefly **dynamics**, is concerned about the relationship between motion of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies (particularly mass and moment of inertia). The foundation of modern-day dynamics is Newtonian mechanics and its reformulation as Lagrangian mechanics and Hamiltonian mechanics.^{[1]}^{[2]}

## History

The field has a long and important history, as remarked by Hamilton:
"The theoretical development of the laws of motion of bodies is a problem of such interest and importance that it has engaged the attention of all the eminent mathematicians since the invention of the dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton." William Rowan Hamilton, 1834 (Transcribed in *Classical Mechanics* by J.R. Taylor, p. 237^{[3]})

Some authors (for example, Taylor (2005)^{[3]} and Greenwood (1997)^{[4]}) include special relativity within classical dynamics.

## Relationship to statics, kinetics, and kinematics

Historically, there were three branches of classical mechanics:

- "statics" (the study of equilibrium and its relation to forces)
- "kinetics" (the study of motion and its relation to forces).
^{[5]} - "kinematics" (dealing with the implications of observed motions without regard for circumstances causing them).
^{[6]}

These three subjects have been connected to *dynamics* in several ways. One approach combined statics and kinetics under the name dynamics, which became the branch dealing with determination of the motion of bodies resulting from the action of specified forces;^{[7]} another approach separated statics, and combined kinetics and kinematics under the rubric dynamics.^{[8]}^{[9]} This approach is common in engineering books on mechanics, and is still in widespread use among mechanicians.

### Fundamental importance in engineering, diminishing emphasis in physics

Today, *dynamics* and *kinematics* continue to be considered the two pillars of classical mechanics. Dynamics is still included in mechanical, aerospace, and other engineering curricula because of its importance in machine design, the design of land, sea, air and space vehicles and other applications. However, few modern physicists concern themselves with an independent treatment of "dynamics" or "kinematics," nevermind "statics" or "kinetics." Instead, the entire undifferentiated subject is referred to as *classical mechanics*. In fact, many undergraduate and graduate text books since mid-20th century on "classical mechanics" lack chapters titled "dynamics" or "kinematics."^{[3]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}^{[16]}^{[17]}
In these books, although the word "dynamics" is used when acceleration is ascribed to a force, the word "kinetics" is never mentioned. However, clear exceptions exist. Prominent examples include *The Feynman Lectures on Physics*.^{[18]}

- List of Fundamental Dynamics Principles

## Axioms and mathematical treatments

- Variational principles and Lagrange's equations
- Hamilton's equations
- Canonical transformations
- Hamilton-Jacobi Theory

## Related engineering branches

- Particle dynamics
- Rigid body dynamics
- Deformation mechanics
- Fluid dynamics

### Related subjects

## References

- ↑ Chris Doran; Anthony N. Lasenby (2003).
*Geometric Algebra for Physicists*. Cambridge University Press. p. 54. ISBN 0-521-48022-1. - ↑ Cornelius Lanczos (1986).
*The variational principles of mechanics*(Reprint of 4th Edition of 1970 ed.). Dover Publications Inc. pp. 5–6. ISBN 0-486-65067-7. - 1 2 3 John Robert Taylor (2005).
*Classical Mechanics*. University Science Books. ISBN 978-1-891389-22-1. - ↑ Donald T Greenwood (1997).
*Classical Mechanics*(Reprint of 1977 ed.). Courier Dover Publications. p. 1. ISBN 0-486-69690-1. - ↑ Thomas Wallace Wright (1896).
*Elements of Mechanics Including Kinematics, Kinetics and Statics: with applications*. E. and F. N. Spon. p. 85. - ↑ Edmund Taylor Whittaker (1988).
*A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies*(Fourth edition of 1936 with foreword by Sir William McCrea ed.). Cambridge University Press. p. Chapter 1, p. 1. ISBN 0-521-35883-3. - ↑ James Gordon MacGregor (1887).
*An Elementary Treatise on Kinematics and Dynamics*. Macmillan. p.*v*. - ↑ Stephen Timoshenko; Donovan Harold Young (1956).
*Engineering mechanics*. McGraw Hill. - ↑ Lakshmana C. Rao; J. Lakshminarasimhan; Raju Sethuraman; Srinivasan M. Sivakumar (2004).
*Engineering mechanics*. PHI Learning Pvt. Ltd. p.*vi*. ISBN 81-203-2189-8. - ↑ David Hestenes (1999).
*New Foundations for Classical Mechanics*. Springer. p. 198. ISBN 0-7923-5514-8. - ↑ R. Douglas Gregory (2006).
*Classical Mechanics: An Undergraduate Text*. Cambridge University Press. ISBN 978-0-521-82678-5. - ↑ Landau, L. D.; Lifshitz, E. M.; Sykes, J.B.; Bell, J. S. (1976). "Mechanics".
**1**. Butterworth-Heinemann. ISBN 978-0-7506-2896-9. - ↑ Jorge Valenzuela José; Eugene Jerome Saletan (1998).
*Classical Dynamics: A Contemporary Approach*. Cambridge University Press. ISBN 978-0-7506-2896-9. - ↑ T. W. B. Kibble, Frank H. Berkshire (2004).
*Classical Mechanics*. Imperial College Press. ISBN 978-1-86094-435-2. - ↑ Walter Greiner; S. Allan Bromley (2003).
*Classical Mechanics: Point Particles and Relativity*. Springer. ISBN 978-0-387-95586-5. - ↑ Gerald Jay Sussman; Jack Wisdom Meinhard; Edwin Mayer (2001).
*Structure and Interpretation of Classical Mechanics*. MIT Press. ISBN 978-0-262-19455-6. - ↑ Harald Iro (2002).
*A Modern Approach to Classical Mechanics*. World Scientific. ISBN 978-981-238-213-9. - ↑ Feynman, RP; Leighton, RB; Sands, M (2003).
*The Feynman Lectures on Physics*. Vol. 1 (Reprint of 1963 lectures ed.). Perseus Books Group. p. Ch. 9*Newton's Laws of Dynamics*. ISBN 0-7382-0930-9.